3A/3B Mathematics Specialist work for Term 3 Week 10
Monday 21 September 2009 -- Concept of the product of vectors
Vector product -- also called cross product -- not covered in this course.
Scalar product -- also called dot product
i.e. the product of the magnitudes multiplied by the cosine of the angle between the vectors (where θ is the angle between the vectors when they are placed head to head or tail to tail). Note that this result is a scalar, not a vector, even though it comes from two vectors.
Properties of the scalar product
Although we talk about this as a product, it is fundamentally different from the products we are accustomed to because we are dealing with vectors. However, many of the properties of the product of numbers also hold for the scalar product. There are also some properties that apply only to the scalar product of vectors.
Commutative property: a.b = b.a
Distributive property:
a.(b+c) = a.b+b.c
(a+b).c) = a.c+b.c
(a+b).(c+d) = a.c+a.d+b.c+b.d
Where vectors are parallel, θ = 0 so cos θ = 1 and a.b = |a||b|.
It follows that a.a=|a|² = a²
Where vectors are perpendicular, θ = 90° so cos θ = 0 and a.b = 0.
Combined with scalar products: a.(λb) = λ(a.b) = (λa).b
a.b is positive for θ acute and negative for θ obtuse (because the vector magnitudes are always positive, the sign of the dot product follows the sign of cosθ).
Note that the associative property does not apply: a.b.c is undefined as the dot product only has meaning when both its arguments are vectors, and whether we group this as (a.b).c or as a.(b.c) we end up with a scalar and a vector so we can't take the dot product.
Indicental notes
There are a number of physical situations where the dot product is directly useful. We will mainly be focusing on its application to minimum distance problems, but it is useful in other situations. For example, you may have encountered the concept of work in your studies in science where it was first defined as force times distance and then later refined to force multiplied by the component of the displacement that is in the direction of the force. This is, in fact, a scalar product. w=f.s=|f||s|cosθ
Work
Work through exercise 8B, referring to the examples in the text where necessary.
Refer to the Sadler Solutions for worked solutions if you have trouble.
Questions
If you get stuck, ask questions here (use the Edit This Page button at the top) or make a post on the Discussion page (see the Discussion tab at the top).
Friday 25 September 2009 -- Scalar Product of component vectors
Finding the scalar product of component vectors is easy: multiply the i components of both vectors, multiply the j components of both vectors, then add the products.
(ai+bj).(ci+dj)=ac+db
Work
Work through exercise 8C, referring to the examples in the text where necessary.
Refer to the Sadler Solutions for worked solutions if you have trouble.
Table of Contents
3A/3B Mathematics Specialist work for Term 3 Week 10
Monday 21 September 2009 -- Concept of the product of vectors
i.e. the product of the magnitudes multiplied by the cosine of the angle between the vectors (where θ is the angle between the vectors when they are placed head to head or tail to tail). Note that this result is a scalar, not a vector, even though it comes from two vectors.
Properties of the scalar product
Although we talk about this as a product, it is fundamentally different from the products we are accustomed to because we are dealing with vectors. However, many of the properties of the product of numbers also hold for the scalar product. There are also some properties that apply only to the scalar product of vectors.Indicental notes
There are a number of physical situations where the dot product is directly useful. We will mainly be focusing on its application to minimum distance problems, but it is useful in other situations. For example, you may have encountered the concept of work in your studies in science where it was first defined as force times distance and then later refined to force multiplied by the component of the displacement that is in the direction of the force. This is, in fact, a scalar product. w=f.s=|f||s|cosθWork
Work through exercise 8B, referring to the examples in the text where necessary.Refer to the Sadler Solutions for worked solutions if you have trouble.
Questions
If you get stuck, ask questions here (use the Edit This Page button at the top) or make a post on the Discussion page (see the Discussion tab at the top).Whiteboard work
Thursday 24 September 2009
No new work introduced. Complete 8A and 8B.Whiteboard work
Friday 25 September 2009 -- Scalar Product of component vectors
Finding the scalar product of component vectors is easy: multiply the i components of both vectors, multiply the j components of both vectors, then add the products.(ai+bj).(ci+dj)=ac+db
Work
Work through exercise 8C, referring to the examples in the text where necessary.Refer to the Sadler Solutions for worked solutions if you have trouble.
Whiteboard work
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